Inserted: 10 dec 2021
Last Updated: 10 dec 2021
Motivated by some models of pattern formation involving an unoriented director field in the plane, we study a family of unoriented counterparts to the Aviles--Giga functional.
We introduce a nonlinear curl operator for such unoriented vector fields as well as a family of even entropies which we call ``trigonometric entropies". Using these tools we show two main theorems which parallel some results in the literature on the classical Aviles--Giga energy. The first is a compactness result for sequences of configurations with uniformly bounded energies. The second is a complete characterization of zero-states, that is, the limit configurations when the energies go to 0. These are Lipschitz continuous away from a locally finite set of points, near which they form either a vortex pattern or a disclination with degree $1/2$. The proof is based on a combination of regularity theory together with techniques coming from the study of the Ginzburg--Landau energy.
Our methods provide alternative proofs in the classical Aviles--Giga context.